I took gains this year from my taxable account I just opened about a year and a half ago from Vanguard. I don't understand why these gains were subtracted from my cost basis. Can someone enlighten me on the mechanics of taxable accounts and changes in cost basis. I am new to these types of accounts as I have always only held most of money in tax deferred.

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It sounds like you may be automatically reinvesting dividends and gains.
Each and every share of a fund that you own has a basis. If you sold shares to "take gains", then the total basis in the fund is reduced by the basis of the shares sold.

Determining the basis of shares sold is probably a different topic.

When you do automatic reinvestment of dividends and gains, it is best to think of a reinvestment transaction as being paid a dividend or gain, and then immediately using that dividend or gain to purchase more shares because that is precisely what is happening. Those new shares have their own basis.

If we are to read this literally, "these gains were subtracted from my cost basis", then it seems that when the sell order was placed it was placed with a "last in first out" directive, or somehow that was the effect of the order placed, and the share price hadn't varied (so the gains were essentially all dividends). Just a guess though. Do you recall how you configured the sell order?

I believe Vanguard is using average cost to figure my basis. I have been taking dividends as income. I just recently sold some shares to add to my income because they were above my initial investment and will be able to get them at 0% for tax purposes because I am in the 15% bracket.

Let's say you bought $10,000 worth over time. Before the sale, say it was worth $12,500. If you sold $2,500, you can't skim the gains off the top and leave the basis as $10,000, if that's what you are thinking. Instead, if they used average cost, you liquidated 1/5 of your holdings, so they would reduce the basis by 1/5 of the original basis, or $2000. So, the sale of $2500 had a basis of $2000 and a gain of $500. Your remaining $10,000 has a basis of $8000.

And if some of the shares were purchased in the last 12 months, that fraction would be a short term capital gains (someone correct me if that's wrong) which you don't want because it'll be at your 15% rate, not 0%.

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